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The Elkies Curve has Rank 28 Subject only to GRH | | Zev Klagsbrun
; Travis Sherman
; James Weigandt
; | | Date: |
23 Jun 2016 | | Abstract: | In 2006, Elkies presented an elliptic curve with 28 independent rational
points. We prove that subject to GRH, this curve has Mordell-Weil rank equal to
28 and analytic rank at most 28. We prove similar results for a previously
unpublished curve of Elkies having rank 27. We also prove that subject to GRH,
certain specific elliptic curves have Mordell-Weil ranks 20, 21, 22, 23, and
24. This complements the work of Jonathan Bober, who proved this claim subject
to both the Birch and Swinnerton-Dyer rank conjecture and GRH. This gives some
new evidence that the Birch and Swinnerton-Dyer rank conjecture holds for
elliptic curves over Q of very high rank. Our results about Mordell-Weil ranks
are proven by computing the 2-ranks of class groups of cubic fields associated
to these elliptic curves. As a consequence, we also succeed in proving that,
subject to GRH, the class group of a particular cubic field has 2-rank equal to
22 and that the class group of a particular totally real cubic field has 2-rank
equal to 20. | | Source: | arXiv, 1606.7178 | | Services: | Forum | Review | PDF | Favorites | |
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